## The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in Plane Co-ordinate Geometry, Proposed at St. John's College, Cambridge, from Dec. 1830 to Dec. 1846. With an Appendix, Containing Several General Properties of Curves of the Second Order, and the Determination of the Magnitude and Position of the Axes of the Conic Section Represented by the General Equation of the Second Degree |

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**bisected**by the cir- cumference . 4. The line joining the centres of the inscribed and cir- cumscribed circles of a triangle subtends at any one of the angular points an angle equal to the semidifference of the other two angles . 5 ... Page 4

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**bisect**it in ; then CP is the diameter , and O the centre of the inner circle PQC when P becomes the point of contact . Let the circle PQC cut AC in Q , and join OQ ; then PQ 2 PA △ POQ = 2 4 PCQ , or ; PO PC ... arc PQ 2PO PC arc PA ... Page 5

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**bisect**DP in E , and draw DF , EF respectively perpendicular to AB , DP meeting each other in F ; with centre F and distance FD describe a circle ; this will pass through the point P and touch the straight line AB ; and △ PDB = the ... Page 8

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**bisected**in C ; and in like manner it may be proved that a , b are the middle points of BC , CA respectively . Draw PM parallel to AC , then if AB , AC be taken for the co - ordinate axes , and x , y be the co- ordinates of P ; PM : AC ... Page 10

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**bisected**by the line joining it with the other extremity . 5. The lines which**bisect**the vertical angles of all triangles on the same base and with the same vertical angle , all intersect in one point . 6. If a semicircle be described ...### Other editions - View all

### Common terms and phrases

2c sin² a₁ a²² ABCD angular points asymptotes axes axis axis-major ay² b₁ bisected C₁ chord of contact co-ordinates conic section conjugate conjugate hyperbola constant cos² cosw curve diagonals diameter draw ellipse equal Euclid find the locus fixed point given points given straight lines hence hk g hyperbola inclined inscribed joining the points latus rectum Let A fig line joining m₁ meet middle point n₂ pair of tangents parabola parallel parallelogram pass through three perpendicular plane points of contact points of intersection polar equation polygon position quadrilateral figure radius remaining sides respectively right angles shew Similarly sin w sin² ST JOHN'S COLLEGE t₁ t₂ tangents be drawn tangents drawn three sides touch vertex y₁

### Popular passages

Page 54 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another ; the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.

Page 117 - Similar triangles are to one another in the duplicate ratio of their homologous sides.

Page 117 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.

Page 96 - The rectangle contained by the diagonals of a quadrilateral ,figure inscribed in a circle, is equal to both the rectangles contained by i'ts opposite sides.

Page 16 - MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another.

Page 28 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one part may be equal to the square on the other part*.

Page 28 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.

Page 10 - ... not in the same plane with the first two ; the first two and the other two shall contain equal angles.

Page 87 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.